Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset

Yilun Shang

doi:10.1109/access.2016.2570518

Abstract

In this paper, the finite-time consensus for arbitrary undirected graphs is discussed. We develop a parametric distributed algorithm as a function of a linear operator defined on the underlying graph and provide a necessary and sufficient condition guaranteeing weighted average consensus in $K$ steps, where $K$ is the number of distinct eigenvalues of the underlying operator. Based on the novel framework of generalized consensus meaning that consensus is reached only by a subset of nodes, we show that the finite-time weighted average consensus can always be reached by a subset corresponding to the non-zero variables of the eigenvector associated with a simple eigenvalue of the operator. It is interesting that the final consensus state is shown to be freely adjustable if a smaller subset of consensus is admitted. Numerical examples, including synthetic and real-world networks, are presented to illustrate the theoretical results. Our approach is inspired by the recent method of successive nulling of eigenvalues by Safavi and Khan.

Highlights

The study of consensus algorithms for multi-agent systems, where distributed processors or agents seek agreement upon a certain quantity of interest via only local information exchange between neighbors, has received a great deal of attention in diverse scientific fields
Shang: Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset linear operator W on an arbitrary undirected graph, and show that average consensus can be reached in K steps with K being the number of distinct eigenvalues of the operator W if and only if W has at least one simple eigenvalue having the eigenvector of all constants
When there are zero entries in the eigenvector associated to the simple eigenvalue, we further prove that finite-time weighted average consensus is reached only by a subset So of nodes corresponding to the non-zero entries

Summary

INTRODUCTION

The study of consensus algorithms for multi-agent systems, where distributed processors or agents seek agreement upon a certain quantity of interest via only local information exchange between neighbors, has received a great deal of attention in diverse scientific fields (see e.g. [1]–[4]). Y. Shang: Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset linear operator W on an arbitrary undirected graph, and show that average consensus can be reached in K steps with K being the number of distinct eigenvalues of the operator W if and only if W has at least one simple eigenvalue having the eigenvector of all constants. Shang: Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset linear operator W on an arbitrary undirected graph, and show that average consensus can be reached in K steps with K being the number of distinct eigenvalues of the operator W if and only if W has at least one simple eigenvalue having the eigenvector of all constants The construction of their algorithm is related to graph filters [20]; see Remark 1.

PROBLEM SETUP

FINITE-TIME GENERALIZED CONSENSUS

NUMERICAL EXAMPLES

CONCLUSIONS